1. Lecture 2

2. Correction Stockinger, - SUSY skript, http://iktp.tu-dresden.de/Lehre/SS2010/SUSY/inhalt/SUSYSkript2010.pdf Drees, Godbole, Roy - "Theory and Phenomenology of Sparticles" - World Scientific, 2004 Baer, Tata - "Weak Scale Supersymmetry" - Cambridge University Press, 2006 Aitchison - "Supersymmetry in Particle Physics. An Elementary Introduction" - Institute of Physics Publishing, Bristol and Philadelphia, 2007 Martin -"A Supersymmetry Primer" hep-ph/9709356 http://zippy.physics.niu.edu/primer.html Unfairly criticised: Now included full superfield chapter (as of 06/09/2011)

3. First lets review what we learned from lecture 1…

4. 1.2 SUSY Algebra (N=1) From the Haag, Lopuszanski and Sohnius extension of the Coleman-Mandula theorem we need to introduce fermionic operators as part of a “graded Lie algebra” or “superalgerba” introduce spinor operators and Weyl representation: Note Q is Majorana (Recap of Lecture 1)

5. Weyl representation: Immediate consequences of SUSY algebra: ) superpartners must have the same mass (unless SUSY is broken). Non-observation ) SUSY breaking (much) Later we will see how superpartner masses are split by (soft) SUSY breaking (Recap of Lecture 1)

6. Weyl representation: (Recap of Lecture 1) Already saw significant consequences of this SUSY algebra: OR

7. Weyl representation: (Recap of Part 1) Already saw significant consequences of this SUSY algebra: decreases spin

8. Extension of electron to SUSY theory, 2 superpartners with spin 0 to electron states We have the states: Electron spin 0 superpartners dubbed ‘selectrons’ The spins of the new states given by the SUSY algebra SUSY chiral supermultiplet with electron + selectron: Take an electron, with m= 0 (good approximation): Simple case (not general solution) for illustration

9. Lecture 2

10. Supersymmetry is a symmetry of the S-matrix. So, So SUSY gives relations between processes involving the pariticles and those with their superpartners. ) Very predictive. SUSY cross-sections 4E

11. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom Proof: Witten index

12. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom Proof: Witten index

13. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom swap Proof: Witten index

14. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom swap Proof: Witten index

15. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom swap Proof: Witten index

16. Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom swap Proof: Where we have used completeness of the set,, twice on the second term in lines 2 & 3 Note: proof assumes and may not be true in the ground state if SUSY is unbroken Witten index

17. Weyl representation: Recall SUSY algebra lead to: 2 states from SM fermion: 2 bosonic states Electron spin 0 superpartners dubbed ‘selectrons’

18. Superpartners Analogously for a scalar boson, e.g. the Higgs, h, has a fermion partner state with either and a gauge boson with s = 1, -1, has a partner majorana fermion as superpartner Higgs, h, with Higgsino with Fermions Sfermions with Vector bosons Gauginos with Warning: Hand waving (details later)

19. 2. SUSY Lagrange density How do we write down the most general SUSY invariant Lagrangian? – construct using two component Weyl spinors, by examining the transformations of scalars, fermions and gauge boson Brute force (See Steve Martin’s primer or Aitchison)* superfields/ superspace – work in a simpler formalism which treats the supersymmetry as an extension of spacetime and superpartners as components of a superfield. (Drees et al, Baer & Tata, our lectures) *Martin now has a full chapter on superfields where he contructs the Lagrangian in a similar way to us, but maintains the brute force approach in earlier chapters

20. 2.2 Superspace Lorentz transformations act on Minkowski space-time: In supersymmetric extensions of Minkowki space-time, SUSY transformations act on a superspace: 8 coordinates, 4 space time, 4 fermionic Grassmann numbers

21. Notational aside: 4 –component Dirac spinors to 2-component Weyl spinors Dirac spinor 2 component Weyl spinors Under Lorentz transformation Form representaions of lorentz group and Left handed Weyl spinor Right handed Weyl spinor

22. Notational aside: 4 –component Dirac spinors to 2-component Weyl spinors Under Lorentz transformation Form representaions of lorentz group and 2 component Weyl spinors Right handed spinor Left handed spinor

23. Dirac spinor2 component Weyl spinors We define: Note Bilinears Lorentz scalar Warning: take care with signs!

24. Bilinears Lorentz scalar Dirac spinor2 component Weyl spinors Warning: take care with signs!

25. Bilinears Lorentz scalar Dirac spinor2 component Weyl spinors Warning: take care with signs!

26. Bilinears Lorentz scalar Dirac spinor2 component Weyl spinors Warning: take care with signs! Home Exercise: prove identities! Further Identities

27. Dirac spinor2 component Weyl spinors Right handed spinor Left handed spinor

28. Dirac spinor2 component Weyl spinors Right handed spinor Left handed spinor

29. Dirac spinor 2 component Weyl spinors Right handed spinor Left handed spinor

30. Dirac spinor2 component Weyl spinors Right handed spinor Left handed spinor For Majorana spinor:

31. Grassmann Numbers Anti-commuting “c-numbers” {complex numbers } If{Grassmann numbers} then Similarly Differentiation: Integration: